Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth

TL;DR

This paper analyzes the asymptotic behavior of a phase field system related to tumor growth as viscosity parameters vanish, providing error estimates and conditions for the uniqueness of the limit system.

Contribution

It extends previous results by allowing two viscosity parameters to tend to zero independently and weakens initial data conditions while maintaining general nonlinearities.

Findings

Proved error estimates for the vanishing viscosity limit.

Established uniqueness of the limit system under growth conditions.

Extended asymptotic analysis to more general initial data and nonlinearities.

Abstract

In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn-Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in the preprint arXiv:1401.5943, letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.